The effect of $S$-accr on intermediate rings between certain pairs of rings

Abstract

The rings considered in this article are commutative with identity and the modules are assumed to be unitary. If $R$ is a subring of a ring $T$, then it is assumed that $R$ contains the identity element of $T$. Let $S$ be a multiplicatively closed subset (m.c. subset) of a ring $R$. In this paper, we consider the concept of $S$-accr, the generalization by Hamed and Hizem of the notion of (accr) in module theory given by Lu. We say that $R$ satisfies (accr) if the increasing sequence of residuals of the form $(I:_{R}B)\subseteq (I:_{R}B^{2})\subseteq (I:_{R}B^{3})\subseteq \cdots$ is stationary for any ideal $I$ of $R$ and for any finitely generated ideal $B$ of $R$. Focusing on certain pairs of rings $R\subseteq T$, the aim of this paper is to study whether $S$-accr on each intermediate ring $A$ between $R$ and $T$ for a suitable m.c. subset $S$ of $A$ (depending on $A$) implies that $A$ satisfies (accr) for each such $A$.